Discretization of Stochastic Differential Systems With Singular - PowerPoint PPT Presentation

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Discretization of Stochastic Differential Systems With Singular Coefficients Part II Denis Talay, INRIA Sophia Antipolis joint works with Mireille Bossy, Nicolas Champagnat, Sylvain Maire, Miguel Martinez, Nicolas Perrin ICERM - Brown – November 2012

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol On PDEs driven by divergence form operators Consider elliptic or parabolic PDEs driven by the strongly elliptic divergence form operator L := 1 2 div( a ( x ) ∇ ) , where 0 < λ | ξ | 2 ≤ ( a ( x ) ξ, ξ ) ≤ Λ | ξ | 2 < + ∞ for all x , ξ ∈ R d .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Techniques related to the generation of semigroups in H 1 ( R d ): • Variational formulations: Aronson, Stroock. • Dirichlet form theory applied to forms of the type E ( u , u ) := 1 � ∇ u ( x ) · a ( x ) ∇ u ( x ) q ( x ) dx , 2 where q is a strictly positive density. • ‘Pseudo SDEs’ (Lyons–Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in W 1 , loc ( R d ), there exists p a pair of local martingales ( M φ , N φ ) respectively adapted with respect to the filtration generated by ( X t , 0 ≤ t ≤ T ) and the filtration generated by ( X T − t , 0 ≤ t ≤ T ), such that � t φ ( X t ) = φ ( X 0 )+1 t +1 t +1 a ( X θ ) ∇ p ( θ, x , X θ ) 2 M φ 2 N φ ·∇ φ ( X θ ) d θ, 2 p ( θ, x , X θ ) 0 and � t � t � M φ � t = a ∇ φ ·∇ φ ( X θ ) d θ and � N φ � t = a ∇ φ ·∇ φ ( X T − θ ) d θ. 0 0 • . . .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Techniques related to the generation of semigroups in H 1 ( R d ): • Variational formulations: Aronson, Stroock. • Dirichlet form theory applied to forms of the type E ( u , u ) := 1 � ∇ u ( x ) · a ( x ) ∇ u ( x ) q ( x ) dx , 2 where q is a strictly positive density. • ‘Pseudo SDEs’ (Lyons–Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in W 1 , loc ( R d ), there exists p a pair of local martingales ( M φ , N φ ) respectively adapted with respect to the filtration generated by ( X t , 0 ≤ t ≤ T ) and the filtration generated by ( X T − t , 0 ≤ t ≤ T ), such that � t φ ( X t ) = φ ( X 0 )+1 t +1 t +1 a ( X θ ) ∇ p ( θ, x , X θ ) 2 M φ 2 N φ ·∇ φ ( X θ ) d θ, 2 p ( θ, x , X θ ) 0 and � t � t � M φ � t = a ∇ φ ·∇ φ ( X θ ) d θ and � N φ � t = a ∇ φ ·∇ φ ( X T − θ ) d θ. 0 0 • . . .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Techniques related to the generation of semigroups in H 1 ( R d ): • Variational formulations: Aronson, Stroock. • Dirichlet form theory applied to forms of the type E ( u , u ) := 1 � ∇ u ( x ) · a ( x ) ∇ u ( x ) q ( x ) dx , 2 where q is a strictly positive density. • ‘Pseudo SDEs’ (Lyons–Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in W 1 , loc ( R d ), there exists p a pair of local martingales ( M φ , N φ ) respectively adapted with respect to the filtration generated by ( X t , 0 ≤ t ≤ T ) and the filtration generated by ( X T − t , 0 ≤ t ≤ T ), such that � t φ ( X t ) = φ ( X 0 )+1 t +1 t +1 a ( X θ ) ∇ p ( θ, x , X θ ) 2 M φ 2 N φ ·∇ φ ( X θ ) d θ, 2 p ( θ, x , X θ ) 0 and � t � t � M φ � t = a ∇ φ ·∇ φ ( X θ ) d θ and � N φ � t = a ∇ φ ·∇ φ ( X T − θ ) d θ. 0 0 • . . .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Techniques related to the generation of semigroups in H 1 ( R d ): • Variational formulations: Aronson, Stroock. • Dirichlet form theory applied to forms of the type E ( u , u ) := 1 � ∇ u ( x ) · a ( x ) ∇ u ( x ) q ( x ) dx , 2 where q is a strictly positive density. • ‘Pseudo SDEs’ (Lyons–Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in W 1 , loc ( R d ), there exists p a pair of local martingales ( M φ , N φ ) respectively adapted with respect to the filtration generated by ( X t , 0 ≤ t ≤ T ) and the filtration generated by ( X T − t , 0 ≤ t ≤ T ), such that � t φ ( X t ) = φ ( X 0 )+1 t +1 t +1 a ( X θ ) ∇ p ( θ, x , X θ ) 2 M φ 2 N φ ·∇ φ ( X θ ) d θ, 2 p ( θ, x , X θ ) 0 and � t � t � M φ � t = a ∇ φ ·∇ φ ( X θ ) d θ and � N φ � t = a ∇ φ ·∇ φ ( X T − θ ) d θ. 0 0 • . . .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Remark. • Pardoux–Williams have exhibited a Lyons–Zheng decomposition for Dirichlet forms with degenerate Neumann boundary conditions. • The Lyons–Zheng decompositions cannot lead to algorithms since one should first compute the transition density p ( t , x , y ) of the Markov process, that is, the fundamental solution .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Remark. • Pardoux–Williams have exhibited a Lyons–Zheng decomposition for Dirichlet forms with degenerate Neumann boundary conditions. • The Lyons–Zheng decompositions cannot lead to algorithms since one should first compute the transition density p ( t , x , y ) of the Markov process, that is, the fundamental solution .

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Parabolic diffraction problems Given a finite time horizon T and a positive matrix-valued function a ( x ) which is smooth except at the interface surfaces between subdomains of R d , consider the parabolic diffraction problem ∂ t u ( t , x ) − 1  2div( a ( x ) ∇ ) u ( t , x ) = 0 for all ( t , x ) ∈ (0 , T ] × R d ,    u (0 , x ) = f ( x ) for all x ∈ R d ,   Compatibility transmission conditions along the interfaces surfaces .  Suppose that 1 2 div( a ( x ) ∇ ) is a strongly elliptic operator. Existence and uniqueness of continuous solutions with possibly discontinuous derivatives along the surfaces hold true: see, e.g. Ladyzenskaya et al. Motivations: Neurosciences (3D brains!), Molecular Dynamics, Geophysics, etc.

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol We consider the one dimensional parabolic problem  ∂ t u ( t , x ) − 1 2 ∂ x ( a ( x ) ∂ x u ( t , x )) = 0 , ( t , x ) ∈ (0 , T ] × ( R − { 0 } ) ,    u ( t , 0+) = u ( t , 0 − ) , t ∈ [0 , T ] ,  ( ⋄ ) u (0 , x ) = f ( x ) , x ∈ R ,    a (0+) ∂ x u ( t , 0+) = a (0 − ) ∂ x u ( t , 0 − ) , t ∈ [0 , T ] . ( ⋆ )  Suppose ∃ λ > 0 , Λ > 0 , 0 < λ ≤ a ( x ) = ( σ ( x )) 2 ≤ Λ < + ∞ for all x ∈ R . Suppose also that σ is of class C 3 b ( R − { 0 } ) and is left and right continuous at point 0. Suppose finally that the first derivative of the function σ has finite left and right limits at 0.

Introduction The one dimensional case The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics The semi-linear 3D Poisson-Boltzmann PDE in Mol The key SDE with weighted local time The one dimensional case allows specific analytical and numerical tools: Portenko (1979), Le Gall (1985), Lejay-Martinez (2003). . . Consider the one-dimensional stochastic differential equation with local time − ( X t ) dt + σ 2 (0+) − σ 2 (0 − ) dX t = σ ( X t ) dB t + σ ( X t ) σ ′ dL 0 t ( X ) . 2 σ 2 (0+) Here L 0 t ( X ) is the right-sided local time corresponding to the sign function defined as sgn( x ) := 1 for x > 0 and sgn( x ) := − 1 for x ≤ 0 and σ ′ − is the left derivative of σ . Under mild hypotheses on σ this SDE has a unique weak solution which is a strong Markov process : see, e.g., Le Gall (1984).